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The Zilber-Pink conjecture for abelian varieties over number fields

Let AA be an abelian variety over a number field and X⊆AQˉX\subseteq A_{\bar{\mathbb Q}} an irreducible subvariety, and let SS be the smallest torsion coset containing XX. A component YY of X∩TX\cap T, for a torsion coset T⊆ST\subseteq S, is atypical if dim⁡Y>dim⁡X+dim⁡T−dim⁡S\dim Y>\dim X+\dim T-\dim S. Zilber (2002, for semiabelian varieties) and Pink (2005) conjectured that such unlikely intersections are controlled, generalizing Manin-Mumford and Mordell-Lang. The curve case was proved by Habegger and Pila (2016); higher dimensions were known only under extra quotient-dimension or height conditions or in codimension one or two. Does every such XX have only finitely many maximal atypical subvarieties?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Diophantine geometry; unlikely intersections
Posed by
Boris Zilber (J. London Math. Soc., 2002) and Richard Pink (ETH preprint, 2005, Conjecture 5.1 for semiabelian varieties)
Year posed
2002
Years open
24y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims that for any abelian variety over a number field, every irreducible subvariety has only finitely many maximal atypical subvarieties inside its smallest containing torsion coset, via a non-density theorem for X∩A[dim⁡X+1]X\cap A^{[\dim X+1]} for XX in no proper torsion coset. A corollary gives the abelian algebraic-point case of Pink's Conjecture 5.2 for finite-rank translates. It does NOT treat semiabelian varieties, abelian varieties over C\mathbb C not defined over Qˉ\bar{\mathbb Q}, or Shimura varieties, and gives no effective or uniform bound.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. The manuscript is credited to OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Theorem 1.1 was read against the posed conjecture: for every abelian variety over a number field and every irreducible subvariety, finitely many maximal atypical subvarieties relative to the smallest containing torsion coset, with no simplicity, CM or height hypothesis. It is the abelian case over Qˉ\bar{\mathbb Q} of Zilber-Pink, not the semiabelian, complex-field or Shimura cases. The reduction to a non-density theorem uses Barroero and Dill's optimal-subvariety argument as a cited input; the height part builds on Vojta, Faltings, Remond and Dill. No Lean formalization. The finiteness is not effective.

Sources

Changelog1 change

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